Lie Algebras Of Cartan Type Research Articles

Varieties of Borel subalgebras for the Jacobson–Witt Lie algebras

Abstract Let W ⁢ ( n ) {W(n)} be the Jacobson–Witt algebra over algebraic closed field 𝕂 {\mathbb{K}} with characteristic p > 2 {p>2} . In [K. Ou and B. Shu, Borel subalgebras of restricted Cartan-type Lie algebras, J. Algebra Appl. 21 2022, 11, Paper No. 2250210], we introduced the so-called B-subalgebra of W ⁢ ( n ) {W(n)} , which serves as an analog of the Borel subalgebra of classical Lie algebras. As a sequel, we describe the structure of the variety ℬ {\mathcal{B}} consisting of all B-subalgebras of W ⁢ ( n ) {W(n)} in this paper. This variety presents an analog of the flag variety for classical Lie algebras. It is shown that ℬ {\mathcal{B}} is related to the variety of all full flags in 𝕂 n + 1 {\mathbb{K}^{n+1}} . Additionally, we provide a detailed description of the varieties for W ⁢ ( 1 ) {W(1)} as an illustrative example. With the above setting-up, one may establish the Springer theory and geometric representations for the Jacobson–Witt algebras.

A generalized Abhyankar’s conjecture for simple Lie algebras in characteristic p>5

In the present paper, we study a purely inseparable counterpart of Abhyankar’s conjecture for the affine line in positive characteristic, and prove its validity for all the finite local non-abelian simple group schemes in characteristic $$p>5$$ . The crucial point is how to deal with finite local group schemes which cannot be realized as the Frobenius kernel of a smooth algebraic group. Such group schemes appear as the ones associated with Cartan type Lie algebras. We settle the problem for such Lie algebras by making use of natural gradations or filtrations on them.

On the Semisimple Orbits of Restricted Cartan Type Lie Algebras W, S and H

In this short note, we give a description of semisimple orbits in the restricted Cartan type Lie algebras W, S, H.

Borel subalgebras of restricted Cartan -Type Lie algebras

It is still an open problem to determine the conjugacy classes of Borel subalgebras of non-classical type Lie algebras. In this paper, we prove that there are at least 2 conjugacy classes of Borel subalgebras as well as maximal triangulable subalgebras of restricted Cartan type Lie algebras of type W, S and H. We are particularly interested in maximal triangulable subalgebras of [Formula: see text] under some conditions which is called [Formula: see text]-subalgebras (Definition 3.1). We classify the conjugacy classes of [Formula: see text]-subalgebras for [Formula: see text] and determine their representatives. This paper and its sequel [Z. Lin, K. Ou and B. Shu, Geometric Setting of Jacobson–Witt Algebras, preprint] attempt to establish both algebraic and geometric setting for geometric representation theory of [Formula: see text]

The Global Sections of Chiral de Rham Complexes on Compact Ricci-Flat Kähler Manifolds

The space of the global sections of the chiral de Rham complex on a compact Ricci-flat Kähler manifold is calculated and it is expressed as the subspace of invariant elements in a \(\beta \gamma -bc\) system under the action of certain Lie algebra of Cartan type.

Generic Property and Conjugacy Classes of Homogeneous Borel Subalgebras of Restricted Lie Algebras of Cartan Type (I): Type W

Generic Property and Conjugacy Classes of Homogeneous Borel Subalgebras of Restricted Lie Algebras of Cartan Type (I): Type W

Simple 14-Dimensional Lie Algebras in Characteristic Two

Using theory of deformations of Lie algebra of type G2, isomorphisms between the known simple 14-dimensional Lie algebras over a field of even characteristic and Lie algebras of Cartan type of S or H are constructed.

Biderivations and linear commuting maps on the restricted Cartan-type Lie algebras H(n; 1)

Let be a field of characteristic p > 3 and be the restricted Cartan-type Lie algebras . We prove that each skew-symmetric biderivation of is inner. As applications of the skew-symmetric biderivations, it is shown that commuting mappings on are scalar multiplication mappings. Moreover, we prove that the commuting automorphisms and dertivations of are the identity mappings and zero mappings, respectively.

Irreducible continuous representations of the simple linearly compact n-Lie superalgebra of type S

In the present paper, we classify all irreducible continuous representations of the simple linearly compact [Formula: see text]-Lie superalgebra of type [Formula: see text]. The classification is based on a bijective correspondence between the continuous representations of the [Formula: see text]-Lie algebras [Formula: see text] and continuous representations of the Lie algebra of Cartan type [Formula: see text], on which some two-sided ideal acts trivially.

Biderivations and linear commuting maps on the restricted Cartan-type Lie algebras and

ABSTRACTLet and denote the restricted Cartan-type Lie algebras and in a field of characteristic , respectively. We prove that each anti-symmetric biderivation of and is inner. As applications of the anti-symmetric biderivations, it is shown that commuting maps on and are scalar multiplication maps. Moreover, we prove that the commuting automorphisms and dertivations of and are the identity mappings and zero mappings, respectively.

The group of automorphisms of the Lie algebra of derivations of a polynomial algebra

A short proof is given of Rudakov’s result (announced in [A. N. Rudakov, Subalgebras and automorphisms of Lie algebras of Cartan type, Funktsional. Anal. i Prilozhen. 20(1) (1986) 83–84]), that the group of automorphisms of the Lie algebra [Formula: see text] of derivations of a polynomial algebra [Formula: see text], over a field of characteristic zero is canonically isomorphic to the group of automorphisms of the polynomial algebra [Formula: see text].

Modular quantizations of Lie algebras of Cartan type K via Drinfeld twists of Jordanian type

We construct explicit Drinfel'd twists of Jordanian type for the generalized Cartan type K Lie algebras in characteristic 0 and obtain the corresponding quantizations, especially their integral forms. By making modular reductions including modulo p and modulo p-restrictedness reduction, and base changes, we derive certain modular quantizations of the restricted universal enveloping algebra u(K(2n+1;1_)) for the restricted simple Lie algebra of Cartan type K in characteristic p. They are new families of noncommutative and noncocommutative Hopf algebras of dimension pp2n+1+1 (if 2n+4≢0(modp)) or pp2n+1 (if 2n+4≡0(modp)) over a truncated p-polynomials ring, which also contain the well-known Radford algebras (see [20]) as Hopf subalgebras. Some open questions are proposed.

ON CARTAN INVARIANTS FOR GRADED LIE ALGEBRAS OF CARTAN TYPE

Let L = X(m; n), X ∈ {W, S, H, K}, be a graded simple Lie algebra of Cartan type over an algebraically closed field of characteristic p > 3. Then L is a so-called generalized restricted Lie algebra. Let [Formula: see text] be the primitive p-envelope of L, and G = X(m; 1), a subalgebra of [Formula: see text]. In this paper, a close connection between Cartan invariants for [Formula: see text] and U(G, χ) is established, where χ ∈ L* is extended to be a linear function on [Formula: see text] trivially, and 1 ≤ ht (χ) < p-2+δXW. This reduces the study of projective representations of the generalized restricted Lie algebra L to the one of the corresponding restricted Lie algebra G. As a special case, we recover some results in [Shu and Jiang, On Cartan invariants and blocks of Zassenhaus algebras, Comm. Algebra33(10) (2005) 3619–3630].

The variety of nilpotent elements and invariant polynomial functions on the special algebra Sn

Abstract In the study of the variety of nilpotent elements in a Lie algebra, Premet conjectured that this variety is irreducible for any finite dimensional restricted Lie algebra. In this paper, with the assumption that the ground field is algebraically closed of characteristic p > 3, we confirm this conjecture for the Lie algebras of Cartan type S˜ n and Sn . Moreover, we show that the variety of nilpotent elements in Sn is a complete intersection. Motivated by the proof of the irreducibility, we describe explicitly the ring of invariant polynomial functions on Sn .

Weyl groups for non-classical restricted Lie algebras and the Chevalley Restriction Theorem

Abstract Let ( 𝔤 , [ p ] ) $(\mathfrak {g},[p])$ be a finite-dimensional restricted Lie algebra, defined over an algebraically closed field k of characteristic p > 0. The scheme of tori of maximal dimension of 𝔤 gives rise to a finite group S(𝔤) that coincides with the Weyl group of 𝔤 in case 𝔤 is a Lie algebra of classical type. In this paper, we compute the group S(𝔤) for Lie algebras of Cartan type and provide applications concerning weight space decompositions, the existence of generic tori and polynomial invariants.

Group gradings on restricted Cartan-type Lie algebras

For a given abelian group G, we classify the isomorphism classes of Ggradings on the simple restricted Lie algebras of types W.mI1/ and S.mI1/ for m 2, in terms of numerical and group-theoretical invariants. Our main tool is automorphism group schemes, which we determine for the simple restricted Lie algebras of types S.mI1/ and H.mI1/. The ground field is assumed to be algebraically closed of characteristic p > 3.

Multiplicity-one representations of divergence-free Lie algebras

Divergence-free Lie algebras (also known as the special Lie algebras of Cartan type) are Lie algebras of volume-preserving transformation groups. They are simple in generic case. Dokovic and Zhao found a certain graded generalization of them. In this paper, we classify all the irreducible and indecomposable multiplicity-one modules of the simple generalized divergence-free Lie algebras.

Support varieties of semisimple-character representations for Cartan type Lie algebras

In this paper, we study the support varieties for Lie algebras L of Cartan type. We give some description for the support varieties of any finitedimensional L-module with character χ, whenever the height of the character χ is not too large. And a more concrete computation can be made for a class of modules with semisimple characters.

Blocks and modules for Whittaker pairs

Inspired by recent activities on Whittaker modules over various (Lie) algebras, we describe a general framework for the study of Lie algebra modules locally finite over a subalgebra. As a special case, we obtain a very general set-up for the study of Whittaker modules, which includes, in particular, Lie algebras with triangular decomposition and simple Lie algebras of Cartan type. We describe some basic properties of Whittaker modules, including a block decomposition of the category of Whittaker modules and certain properties of simple Whittaker modules under some rather mild assumptions. We establish a connection between our general set-up and the general set-up of Harish-Chandra subalgebras in the sense of Drozd, Futorny and Ovsienko. For Lie algebras with triangular decomposition, we construct a family of simple Whittaker modules (roughly depending on the choice of a pair of weights in the dual of the Cartan subalgebra), describe their annihilators, and formulate several classification conjectures. In particular, we construct some new simple Whittaker modules for the Virasoro algebra. Finally, we construct a series of simple Whittaker modules for the Lie algebra of derivations of the polynomial algebra, and consider several finite-dimensional examples, where we study the category of Whittaker modules over solvable Lie algebras and their relation to Koszul algebras.

Generators of simple Lie algebras II

This paper is a continuation of earlier work on generators of simple Lie algebras in arbitrary characteristic [3]. We show that, in contrast to classical Lie algebras, simple graded Lie algebras of Cartan type S, H or K never enjoy the “one-and-a-half generation” property. The methods rely on a study of centralisers in Cartan type Lie algebras.